Many technical processes and systems can be described by differential equations DGL. One or more differential equations can be used to model e.g. chemical processes (in particular combustion processes), many-body systems or transport systems. Many-body systems comprise e.g. automobiles, robots or machine tools. Differential equations can also be used to describe or model electrical circuits such as oscillators or digital circuits, for example. Further examples include transport processes and diffusion processes in the case of fluids and gases, or aging processes such as radioactive decay or population growth of e.g. microorganisms and the like.
FIG. 1 schematically shows a technical system to which input parameters K are applied, and which delivers output parameters A. The technical system is modeled or described by means of differential equations DGL. In many technical systems, the input parameters K are stochastic parameters or varying parameters. One example of varying input data or input parameters is the ambient temperature in the case of a chemical system for manufacturing or converting chemical products. Measured data that is delivered by e.g. measuring sensors can also have a stochastic distribution. In addition, e.g. the geometry of a many-body system can vary or deviate from a standard value due to manufacturing tolerances. In general, manufacturing tolerances of components that are present in a system constitute a possible source of variation of input parameters K. Deviations of the stochastic input parameters result in uncertainty in relation to the output parameters A of the system.
For example, a system can be a simulation model of a real technical system that is described by one or more differential equations. A typical example is a climatic model which is used to predict a future temperature profile T(t) for a future time period ΔT as a function of stochastic input parameters K. In the context of this simulation model, the future temperature profile T(t) represents an output parameter A(t), which is determined as a function of stochastic input parameters K. The stochastic input parameters K are e.g. the air pressures that are measured by one or more sensors at different locations. The measured stochastic input parameters K vary within a certain range, e.g. due to the manufacturing tolerances of the various sensors. The various uncertainties in relation to the various stochastic input parameters K proliferate in the calculation of the output parameter A(t), e.g. the predicted temperature profile T(t), such that a temperature prediction becomes more inaccurate as the calculated temperature value moves further into the future.
In the case of conventional simulation methods, such uncertainties (e.g. variation due to manufacturing tolerances) are taken into account using a so-called Monte Carlo evaluation. In this case, a random number set which reflects a probability distribution is generated, and the existing differential equation system is checked thoroughly using the corresponding generated values. This conventional method delivers the moments of the trajectory at the designated end time. This conventional approach results in considerable computing effort, particularly in the case of a complex technical system which has a complex differential equation system comprising a multiplicity of differential equations and a multiplicity of input parameters K. For example, in the case of such a conventional Monte Carlo evaluation of an output differential equation system, several months are required in some cases before the temporal profile of a specific output parameter Δ can be calculated. A microcomputer performing the calculation of an output parameter may require several thousand evaluations before a satisfactory convergence occurs.
Such a significant calculation effort and the associated lengthy calculation duration for the evaluation of an output differential equation system having stochastic input parameters prohibits any use in a corresponding control or regulation of the input parameters, particularly in the case of a technical installation in which input parameters must be adjusted in real time.